I. Field of the Invention
The present invention relates to cellular communications and also relates to the Nyquist rate for data symbol transmission, the Shannon bound on communications capacity, and symbol modulation and demodulation for high-data-rate satellite, airborne, wired, wireless, and optical communications and includes all of the communications symbol modulations and the future modulations for single links and multiple access links which include electrical and optical, wired, mobile, point-to-point, point-to-multipoint, multipoint-to-multipoint, cellular, multiple-input multiple-output MIMO, terrestrial networks, and satellite communication networks. In particular it relates to WiFi, WiFi 802.11ac, WiMax, long-term evolution LTE, 3G, 4G for cellular communications and satellite communications. WiFi, WiMax use orthogonal frequency division multiplexing OFDM on both links and LTE uses single carrier OFDM (SC-OFDM) on the uplink from user to base station and OFDM on the downlink form base station to user. WiMax occupies a larger frequency band than WiFi and both use OFDM waveforms. SC-OFDM LTE is a single carrier orthogonal waveform version of OFDM which uses orthogonal frequency subbands of varying widths.
II. Description of the Related Art
Bounds on current communications capacity are the communications Nyquist rate, the Shannon rate, and the Shannon capacity theorem. The Nyquist complex sample rate is 1/T=B where B is the signal bandwidth, the Shannon rate W≧B in Hz is the frequency band W=(1+α)B where α is the excess bandwidth required to capture the spillover of the signal spectrum beyond B with a representative value being α=0.25, and the Shannon capacity theorem specifies the maximum data rate C in Bps (bits/second) which can be supported by the communications link signal-to-noise power ratio SNR=S/N over W.
The Nyquist rate 1/T is the complex digital sampling rate 1/T=B that is sufficient to include all of the information within a frequency band B over a communications link. Faster than Nyquist rate communications (FTN) transmits data symbols at rates 1/Ts≧1/T wherein 1/Ts is the data symbol transmission rate in the frequency band B which means Ts is the spacing between the data symbols. FTN applications assume the communications links with a data symbol rate equal to the Nyquist rate 1/Ts=1/T=B operate as orthogonal signaling with no intersymbol interference (ISI) between the demodulated data symbols. It is common knowledge that the communications data symbol rate 1/Ts for orthogonal signaling can be increased to as high as 25% above the Nyquist complex sample rate 1/Ts=1/T=B with very little loss in Eb/No and with no perceptible loss in some cases using simple data symbol modulations. Above 25% it has been observed that there is a rapid loss in signal strength. Only a few studies have addressed data symbol rates above 25%. An example in U.S. Pat. No. 8,364,704 transmits digital bit streams at FTN rates and depends on the transmit signal alphabets to be in distinct locations on receive to enable alphabet detection. There is no comparison or proof that the data rate performance is comparable to the performance using conventional orthogonal signaling.
The Shannon bound [1]-[5] on the maximum data rate C is complemented by the Shannon coding theorem, and are defined in equations (1).
Shannon bound and coding theorem
1 Shannon capacity theoremC=W log2(1+SNR)  (1)
2 Shannon coding theorem for the information bit rate Rb                 For Rb<C there exists codes which support reliable communications        For Rb>C there are no codes which support reliable communicationswherein C in Bps is the channel capacity for an additive white Gaussian noise AWGN channel in W, “log2” is the logarithm to the base 2, and C is the maximum rate at which information can be reliably transmitted over a noisy channel where SNR=S/N is the signal-to-noise ratio in W.        
MIMO communications enable higher capacities to be supported with multiple independent links over the same bandwidth. This multiple-input multiple-output MIMO requires the physical existence of un-correlated multiple communications paths between a transmitter and a receiver. MIMO uses these multiple paths for independent transmissions when the transmission matrix specifying these paths has a rank and determinant sufficiently large to support the paths being used. In MIMO U.S. Pat. No. 7,680,211 a method is disclosed for constructing architectures for multiple input transmit and multiple output receive MIMO systems with generalized orthogonal space-time codes (C0) which are generalization of space-time codes C and generalizations (H0) of the transmission matrix (H) that enable the MIMO equation Y=Hf(C,X)+No to be written Y=H0C0X+No which factors out the input signal symbol vector X and allows a direct maximum-likelihood ML calculation of the estimate {circumflex over (X)} of X, and wherein Y is the received (Rx) symbol vector, No is the Rx noise vector, and f(C,X) is a non-separable encoding C of X.
OFDM waveform implement the inverse FFT (IFFT=FFT−1) to generate OFDM (or equivalently OFDMA which is orthogonal frequency division multiple access to emphasize the multiple access applications). OFDM uses pulse waveforms in time and relies on the OFDM tone modulation to provide orthogonality. SC-OFDM is a pulse-shaped OFDM that uses shaped waveforms in time to roll-off the spectrum of the waveform between adjacent channels to provide orthogonality, allows the user to occupy subbands of differing widths, and uses a different tone spacing, data packet length, and sub-frame length compared to OFDM for WiFi, WiMax. In addition to these applications the symbol modulations 4PSK, 8PSK, 16QAM, 64QAM, 256QAM are used for satellite, terrestrial, optical, and nearly all communication links and with maximum data symbol rates achieved using 256QAM.